1520s, the older name of the nursery play known in U.S. as peek-a-boo; see peep (n.1). The nursery rhyme seems to date to c. 1800.
Entries linking to bo-peep
peep n.1
"a furtive look, as if through a crevice, a glimpse," 1520s, originally and especially "the first looking out of light from the eastern horizon" (the sense in peep of day); from peep (v.1). General meaning "a furtive glance" is attested by 1730.
peekaboo n.
also peek-a-boo, as the name of a children's game attested from 1590s; as an adjective (of garments) meaning "see-through, open," it dates from 1895. From peek (v.) + boo. Compare bo-peep.